Automata operating on pairs of words were introduced as an alternative way of capturing acceptance of regular $\omega$-languages. Families of DFAs and lasso automata operating on such pairs followed, giving rise to minimisation algorithms, a Myhill-Nerode theorem and language learning algorithms. Yet Kleene theorems for such a well-established class are still missing. Here, we introduce rational lasso languages and expressions, show a Kleene theorem for lasso languages and explore the connection between rational lasso and $\omega$-expressions, which yields a Kleene theorem for $\omega$-languages with respect to saturated lasso automata. For one direction of the Kleene theorems, we also provide a Brzozowski construction for lasso automata from rational lasso expressions.

中文翻译：

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Lasso 语言和 $ω$-语言的 Kleene 定理

引入对单词对进行操作的自动机作为捕获常规 $\omega$ 语言接受度的替代方法。随后出现了在此类对上运行的 DFA 和套索自动机系列，从而产生了最小化算法、Myhill-Nerode 定理和语言学习算法。然而，这样一个完善的类别的克莱恩定理仍然缺失。在这里，我们介绍有理套索语言和表达式，展示套索语言的克林定理，并探讨有理套索和 $\omega$ 表达式之间的联系，这产生了 $\omega$-语言关于饱和套索自动机的克林定理。对于 Kleene 定理的一个方向，我们还根据有理套索表达式提供了套索自动机的 Brzozowski 构造。